Passive Control of a Vibrating Bar with a Dynamic Vibration Absorber

International Journal of Innovative Research in Advanced Engineering (IJIRAE) ISSN: 2349-2763 Issue 08, Volume 3 (August 2016) www.ijirae.com

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Passive Control of a Vibrating Bar with a Dynamic

Vibration Absorber

Giman KIM* Department of Mechanical System Engineering,

Kumoh National Institute of Technology, South Korea.

Abstract — The passive control of a vibrating bar is analyzed in this paper. It has been controlled passively by a Dynamic Vibration Absorber (DVA) which consists of a mass and a spring. A bar, which is fixed at one end and free at the other end is subjected to a harmonically excited force being positioned at the free end. By analyzing the motion of a bar, the equation of motion for a bar was derived by using a method of variation of parameters. To define the optimal conditions of a DVA, the cost function, which denotes the reduction of a force being transmitted to the fixed end of a bar was evaluated and discussed. The possibility of reduction of the force transmissibility was found to depend on the location and the spring stiffness of a DVA.

Keywords— Passive control, a method of variation of parameters, the force transmissibility, a Dynamic Vibration Absorber (DVA), spring stiffness

I. INTRODUCTION

In the industrial fields, vibrations of machines and structures had been a long standing problem to be solved. So far a lot of studies have been processed to have good results from the vibration control skills in machining operations in the past decades. Currently as of the optimal design condition, the fast manufacturing time and the accurate level of the machining center and mother machine are known to be the important factors. In the high speed operation, the vibration which may occurs at each parts of machine should be controlled actively or passively for the stability of the machine structure.

For the simplification of the complex structures, each part of the machines is considered as a continuous system such as bar, beam, plate or shell. Hence, the vibration problems of a continuous system had been studied by lots of researchers. Recently the vibration energy flow and the dynamic response of a beam, plate, shell or some compound system have been analyzed. By using a model of an elastic beam, the vibration energy and control technologies had been presented [1-3]. Furthermore, the complex frame which is the assembly of several beams had been used to analyze and control the vibration characteristics [4-5]. The Active control method had been proven to become the convenient control method for the known forcing frequency zone [6-8]. The passive control of the vibrations of the machining center and machine tools had been introduced [9-12].

A working arm or part in machinery is known to experience the longitudinal vibration being induced by the constant operation speed. The Dynamic Vibration Absorber (DVA) is known to be effective over narrow band of frequencies and is designed to make the natural frequencies of the main system be away from the forcing frequency. Hence in this study a theoretical system which consists of a Dynamic Vibration Absorber and a bar with an external force at the free end is employed to evaluate and reduce the force transmissibility of a bar. The edges condition of a bar is fixed at one end and free at the other end. Based on the wave equation, the vibration of a bar will be discussed. To define the optimal conditions for the passive control of the bar, the cost function which is the ratio of the force transmissibility of a bar without DVA to that of a bar with DVA will be discussed. Fig. 1 shows the theoretical model of a uniform bar which is subjected to a harmonically excited force F(t) at the free end and has a Dynamic Vibration Absorber (DVA) which is located at a distance of xa from the left end of the bar.



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Fig. 1: Theoretical model of a bar with fixed and free ends

F(t) (the external force), Dynamic Vibration Absorber (DVA)[mass (ma), spring stiffness (k), location (Xa)], BAR PROPERTIES [LENGTH (L), DENSITY (), CONSTANT SECTION AREA (A), YOUNG’S MODULUS (E)]

II. THE GOVERNING EQUATIONS

A uniform bar of a cross section area A, mass density and length L, which is fixed at the left end and free at the right end is subjected to a harmonically excited force F(t) at the free end and has a Dynamic Vibration Absorber (DVA) which is located at a distance of xa from the left end of the bar. The governing equations of a vibrating bar and a Dynamic Vibration Absorber (DVA) can be written as:

axxtxutyk

t

txuA

x

txuA ),()(2

),(2 2

),(2E (1)

),()(2)(2

axxtxkutkyt

tyma

(2)

Where u(x,t) and y(t) represent the displacement of a bar and a DVA, respectively. In the above equations, the displacements of a bar and a DVA are given as the time harmonic function. Hence the displacements (u(x,t) and y(t)) are expressed into the terms of the time harmonic motion as,

Y and )(u),( titi ey(t)extxu (3) Where u(x) and Y mean the deflection of a bar and a DVA, respectively and is the circular frequency. By inserting Eq. (3) into Eq. (1) and Eq. (2), then suppressing the time term, Eq.(1) and Eq. (2) can be rearranged as,

)()(u)(u )(u 22

2

axxxkkYxAx

xEA

(4)

)(u2

ak-m

xxxkY aa

(5)

Then by using Eq. (5), Eq. (4) can be rewritten as

)()(u)(u )(u2

22

2

2

aaa

a xxxmkEA

kmxx

x

(6)

Where the wave number =2/E. The boundary conditions of a bar with fixed and free ends are given as

tio

Lxx

efx

Eu

uA and 0 0

(7)

Where fo is the magnitude of the external force. The total solution of Eq. (6) can be expressed into the sum of a homogeneous solution (uh) and a particular solution (up) as

)(u)(u)(u xxx ph (8) The homogeneous solution is determined by letting the right side of Eq. (6) be zero and then becomes as

xAxAxh cossin)(u 21 (9)

Here the particular solution can be solved by means of a method of variation of parameters and then be assumed as




xxVxxVxp cos)(sin)()(u 21

(10) Where the coefficients V1 and V2 are can be determined by means of the method of variation of parameters and defined as

follows

dfk

xVanddfk

xVxx

sin1)(cos1)( 21 (11)

).()(u, 2

2

aapa

a xxmkEA

kmffunctionforcingthewhere

The final form of the particular solution can be obtained by using Eqs. (10) and (11). The complete solution of Eq. (8) can be written as

aaaa

a

aaaa

a

xxHxxxmkEA

kmxA

xxHxxxmkEA

kmxAx

sincoscos

sinsinsin)(u

2

2

2

2

2

1

(12)

Where H(x) represents the Heaviside unit step. By using the boundary conditions given in Eq. (7), the final solution of the dynamic response of a bar becomes as,

aaaa

a xxHxxxmkEA

kmxAx

sinsinsin)(u 2

2

1 (13)

Where A2 becomes zero and the constant function A1 is as follows,

aaa

a

o

xLxmkEA

kmLEA

fA

cossincos 2

21 (14)

Here Eq. (13) introduces the deflection of a bar with a DVA, which is caused by the harmonically excited force applied to the free end.

III. NUMERICAL RESULTS AND DISCUSSION

All values obtained in this study have been expressed into the nondimensional forms. In Table 1, the nondimensional properties and the given values are introduced as

TABLE I - NONDIMENSIONAL PROPERTIES AND THE GIVEN VALUES EXPRESSION FORM VALUE A MASS RATIO M = MA/ L 0 ~ 0.3 EXTERNAL FORCE RATIO FO=FO/EA 1 A STRUCTURAL DAMPING 0.001 A DVA LOCATION XA=XA/L 0.1 ~ 0.9 A STIFFNESS RATIO KS=K/(EA/L) 0.001 ~ 10

A. Resonance Frequency Coefficients()

In Eq. (14), the denominator part vanishes for certain specific values of the wave number . The roots of denominator which are expressed in r by 2=2/E are called the resonance frequencies for the bar with a DVA. So the resonance frequency equation becomes as,

0cossincos)( 2

2

ooa

a xLxmkEA

kmLF

(15)

Here Eq. (15) can be rewritten in terms of the nondimensional form as,

01cossin1

sin)( 2

aa

s

XX

Km

mF

(16)

where = L. Equation (16) is a transcendental equation in and its roots must be obtained numerically. For two stiffness ratios (Ks = 0.01 and 1), the variations of the first resonance frequency coefficient () versus location of DVA for three cases of mass ratio (m = 0.1, 0.2, 0.3) were plotted in Fig. 2. For the low stiffness (Ks = 0.01) of DVA in Fig. 2 (A), the variation of the first resonance frequency coefficient () is found to be varied small along the location of a DVA but not sensitive to the mass of a DVA.




However in case of the high stiffness (Ks = 1) of DVA in Fig. 2 (B), the variation of the first resonance frequency coefficient () is found to be very sensitive to the location and mass of a DVA. It is noted that as the mass of a DVA is getting larger, the values of the first resonance frequency coefficient () are getting smaller. In Fig. 2(A), as the location of a DVA is close to the free end, the values of the first resonance frequency coefficient () are increasing gradually. But in Fig. 2(B), as the location of a DVA is close to the free end, the values of the first resonance frequency coefficient () are decreasing gradually. So it is noted that the DVA with a low stiffness gives the additional stiffness effect to the behaviour of the vibrating bar and the DVA with a high stiffness gives both effects of the additional mass and stiffness to the behaviour of the vibrating bar. In Fig. 3, the above mentioned notations might be proved. For two mass ratios (m = 0.1 and 0.3), the variations of the first resonance frequency coefficient () versus location of DVA for four cases of stiffness ratio (Ks = 0.01, 0.1, 1, 3) were plotted in Fig. 3. In cases of the low stiffness (Ks = 0.01, 0.1), the values of is increasing a little bit along the further distance from the fixed end of a bar. But in cases of the high stiffness (Ks = 1, 3), the values of is decreasing along the further distance from the fixed end of a bar.

(A) (B)

Fig. 2 Resonance frequency coefficient (1) vs. location of DVA : (A) Ks = 0.01 and (B) Ks = 1

(A) (B)

Fig. 3 Resonance frequency coefficient (1) vs. location of DVA : (A) m = 0.1 and (B) m = 0.3

B. Reduction of The Force Transmissibility (Tf) The force transmissibility (Tf) of a vibrating bar is expressed as follows;

0.0 0.2 0.4 0.6 0.8 1.01.50

1.52

1.54

1.56

1.58

1.60

Location of DVA (Xa)

stiffness ratio(ks) = 0.01

Freq

uenc

y co

effic

ient

()

m=0.1 m=0.2 m=0.3

0.0 0.2 0.4 0.6 0.8 1.01.0

1.2

1.4

1.6

1.8

2.0


m=0.1 m=0.2 m=0.3

stiffness ratio(ks) = 1

Freq

uenc

y co

effic

ient

()

0.0 0.2 0.4 0.6 0.8 1.01.0

1.2

1.4

1.6

1.8

2.0


mass ratio(m) = 0.1

Freq

uenc

y co

effic

ient

()

Ks = 0.01 Ks = 0.1 Ks = 1 Ks = 3

0.0 0.2 0.4 0.6 0.8 1.01.0

1.2

1.4

1.6

1.8

2.0

Ks = 0.01 K

s = 0.1

Ks = 1

Ks = 3


mass ratio(m) = 0.3

Freq

uenc

y co

effic

ient

()




0T

T

u F

x

of

xEAwhere

FFT

(17)

In Eq. (17), the structural damping (= 0.001) was introduced to avoid being infinite at the resonance. The reduction level of the force transmissibility (Tf) was evaluated in comparison with Tf of a bar without DVA and Tf of a bar with DVA. The ratio of Tf of a bar without DVA to Tf of a bar with DVA is expressed in terms of decibel [dB] as follows;

DVAf

noDVAff T

TLogT

]Re[]Re[

10[dB] of Reduction 10 (18)

where ‘Re’ means the real part taking from the complex number of Tf. For the two arbitrary values of the stiffness (Ks = 0.1 and 2) ratio, Figs. 4 (A) and (B) shows the reductions of the force transmissibility (Tf) of two locations (Xa=0.8 and 0.2) of DVA versus the forcing frequency coefficient (f), respectively. As shown in Fig. 4, the levels of the reduction of Tf were observed to be good or bad irregularly along the forcing frequency coefficient (f). It means that a DVA with the arbitrary value of stiffness may not apply to a main system which is subjected to any forcing frequency.

(A) (B)

Fig. 4 Reduction of Tf vs. forcing frequency coefficient (f); m=0.2, Solid(Ks=0.1), Dot(Ks=2) (A) Xa=0.8 and (B) Xa=0.2

According to the previous study [13], the optimal design condition of a DVA was reported such that the natural frequency of DVA should be set equal to the forcing frequency of a main system. So the stiffness ratio can be given as;

2s foptimal

mK (19)

Where f is the forcing frequency coefficient. For the given forcing frequency coefficient, the stiffness ratio of a DVA can be determined by using Eq. (19).

(A) (B)

Fig. 5 Reduction of Tf vs. Location of DVA (Xa); (A) f=1 and (B) f=5

2 4 6 8 10

-20

-10

10

20

30

dB

2 4 6 8 10

-20

-10

10

20

dB

0.0 0.2 0.4 0.6 0.8 1.0202224

262830323436

3840424446

4850

m = 0.1, Ks=0.1 m = 0.2, Ks=0.2 m = 0.3, Ks=0.3

f = 1


Red

uctio

n of

Tf [

dB]

0.0 0.2 0.4 0.6 0.8 1.02022

2426

28

3032

3436

3840

4244

4648

50

m = 0.1, Ks=2.5 m = 0.2, K

s=5

m = 0.3, Ks=7.5

f = 5


Red

uctio

n of

Tf [

dB]




In Figs. 5 (A) and (B) for f =1 and 5, the reduction of Tf were evaluated along the location of DVA for three cases of mass ratio (m=0.1, 0.2, 0.3). As is the greater mass ratio, the higher reduction is observed. As observed in Fig. 5, the reduction of Tf are obtained over 20 dB at all locations of DVA based on the force transmissibility of a main system without DVA and the effect of DVA’s location exhibits the spatial form of the normal modes corresponding to each natural frequency of the bar (1=1.57 and 2 = 4.71).

In Figs. 6 (A) and (B), the reduction of Tf for four cases of the stiffness ratio are plotted versus location of DVA for two mass ratios (m = 0.1 and 0.2), respectively. As shown in Fig. 6, the reduction loci of four stiffness ratios are very similar in two mass ratios which look like the spatial form of the normal modes of the bar. As stated previously, the reduction levels are greater than 20 dB for the most cases except the low values of the stiffness ratio (Ks = 0.002) which is corresponding to the low forcing frequency coefficient (f = 0.1).

(A) (B) Fig. 6 Reduction of Tf vs. Location of DVA (Xa); (A) m = 0.1 and (B) m = 0.2

IV. CONCLUSIONS

On the bases of the analyses in this study, the conclusions are obtained as follows,

- Passive control of Dynamic Vibration Absorber (DVA) is observed to have the satisfactory results for the reduction of the force transmissibility which is the ratio of the force transmitted to a fixed end to the external force of the bar.

- The location and the stiffness of DVA are known to become the important factor for the control strategy. - At the low stiffness of DVA, the effect of the DVA’s mass is very weak. - It is proven that the optimal stiffness of DVA is proportional to square of the forcing frequency and the mass of DVA.

ACKNOWLEDGMENT

This paper was supported by Research Fund, Kumoh National Institute of Technology.

REFERENCES

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0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

40

45

50

55

60

65

70

f= 0.1, Ks=0.002

f= 1, Ks=0.2

f= 3, Ks=1.8

f= 5, Ks=5


m = 0.1

Red

uctio

n of

Tf [

dB]

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

35

40

45

50

55

60

65

70


m = 0.2

f= 0.1, Ks=0.002

f= 1, Ks=0.2

f= 3, Ks=1.8

f= 5, Ks=5

Red

uctio

n of

Tf [

dB]




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[13] Snowdon, J.C., Vibration and Shock in Damped Mechanical Systems, Wiley, New York, pp 316-331, 1968.

Passive Control of a Vibrating Bar with a Dynamic Vibration Absorber

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